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Energy-efficient forwarding strategies for wireless sensor networks in fading channels

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Abstract

In the context of geographic routing in wireless sensor networks linked by fading communication channels, energy-efficient transmission is important to extend the network lifetime. To this end, we propose a novel method to minimize the energy consumed by one bit of information per meter and per second toward the destination in fading channels. Using the outage probability as a measure to maximize the amount of information delivered within a given time interval we decide energy-efficient geographic routing between admissible nodes in a wireless sensor network. We present three different approaches, the first is optimal and is obtained by varying both transmission rate and power, the other two are sub-optimal since only one of them is tuned. Simulation examples comparing the energy costs for the different strategies illustrate the theoretical analysis in the cases of log-normal and Nakagami shadow fading. With the method proposed it is possible to obtain significant energy savings (up to ten times) with respect to fixed transmission rate and power.

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Author information

Authors and Affiliations

  1. Grupo Control Automático y Sistemas (GCAyS), Universidad Nacional del Comahue, Buenos Aires 1400, 8300, Neuquén, Argentina

    R. Milocco

  2. Institut National de Recherche en Informatique et en Automatique, 2 Rue Simone IFF, 75012, Paris, Francia

    P. Muhlethaler

  3. Conservatoire National des Arts et Metiers, Centre d’Etude et De Recherche en Informatique et Communications (CNAM/CEDRIC), 292 rue Saint-Martin, 75003, Paris, France

    S. Boumerdassi

Authors
  1. R. Milocco
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  2. P. Muhlethaler
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  3. S. Boumerdassi
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Correspondence to R. Milocco.

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Appendices

Appendix 1

\(\mathcal {P}_{o}(v)\) is a continuous and monotonically decreasing function of v from \(\mathcal {P}_{o}(-\infty )=1\) to \(\mathcal {P}_{o}(\infty )=0\). Then, for any arbitrary value v∗ it follows that \(\mathcal {P}_{o}(v^{*})\in (0,1)\), and the derivative in Eq. (20) belongs in the interval \((-\log (10)/10, 0)\). Then, invoking the mean value theorem there exists at least one value \(\overline {v}>v^{*}\) and \(\underline {v}<v^{*}\) such that the following holds:

$$ \begin{array}{@{}rcl@{}} \left.\frac{\partial \mathcal{P}_{o}(v)}{\partial v}\right|_{v=v^{*}} &=&-\frac{\log(10)}{10}(1-\mathcal{P}_{o}(v^{*}) \end{array} $$
(A1.1)
$$ \begin{array}{@{}rcl@{}} &=&\frac{\mathcal{P}_{o}(\overline{v})-\mathcal{P}_{o}(\underline{v})}{\overline{v}-\underline{v}} \end{array} $$
(A1.2)

which proves that there always exists at least one pair \((v, \mathcal {P}_{o}(v))\) that satisfies Eq. (20).

Appendix 2

Denoting by \(\bar {x}\) the mean value of a random variable x, the sub-fix o for PR-control, p for P-control, and r for R-control, from Eq. (12), considering β = 0 for simplicity, and taking into account that in all cases the pair (\(v, \mathcal {P}_{o}\)) is the same, which means the comparison is made over the same fading, we obtain the following mean values for the energy in decibels:

$$ \begin{array}{@{}rcl@{}} \bar{E}_{r}^{dB}&=&P_{r}^{dB} - \bar{R}_{r}^{dB}+\alpha^{dB} - \bar{d}^{dB}, \end{array} $$
(A2.1)
$$ \begin{array}{@{}rcl@{}} \bar{E}_{p}^{dB}&=&\bar{P}_{p}^{dB} - R_{p}^{dB}+\alpha^{dB} - \bar{d}^{dB}. \end{array} $$
(A2.2)

where Rp and Pr are constant values. From Eq. (16) we obtain,

$$ \begin{array}{@{}rcl@{}} P_{r}^{dB}&=&\overline{SNR}_{Rr}^{dB}+ v+\gamma \bar{d}^{dB}+P_{n}, \end{array} $$
(A2.3)
$$ \begin{array}{@{}rcl@{}} \bar{P}_{p}^{dB}&=&SNR_{Rp}^{dB}+ v+\gamma \bar{d}^{dB}+P_{n}. \end{array} $$
(A2.4)

where \(\overline {SNR}_{Rr}=\mathcal {E}[e^{Rr}-1]\) and SNRRp = eRp − 1. Replacing in Eqs. (A2.1) and (A2.2) we obtain,

$$ \begin{array}{@{}rcl@{}} \bar{E}_{r}^{dB}&=&\overline{SNR}_{Rr}^{dB}- \bar{R}_{r}^{dB}+L, \end{array} $$
(A2.5)
$$ \begin{array}{@{}rcl@{}} \bar{E}_{p}^{dB}&=&SNR_{Rp}^{dB} - R_{p}^{dB}+L . \end{array} $$
(A2.6)

where \(L=\alpha ^{dB}+ v+(\gamma -1) \bar {d}^{dB}+P_{n}\) is a constant. Since Rr is a design variable that minimize Er for each value of d, the following property holds: \(\overline {SNR}_{Rr}=\mathcal {E}[e^{Rr}-1]\neq e^{\mathcal {E}[Rr]}-1\). This optimal can never be satisfied by a constant value Rp for which \(\overline {SNR}_{Rp}=\mathcal {E}[e^{Rp}-1]= e^{\mathcal {E}[Rp]}-1\). Thus, always fullfils \(\bar {E}_{r}^{dB}\leq \bar {E}_{p}^{dB}\).

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Milocco, R., Muhlethaler, P. & Boumerdassi, S. Energy-efficient forwarding strategies for wireless sensor networks in fading channels. Ann. Telecommun. 76, 97–108 (2021). https://doi.org/10.1007/s12243-020-00815-x

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